Math Grade 8 Assessment Anchors and Eligible Content

Math Grade 8 Assessment Anchors and Eligible Content Pennsylvania Department of Education www.pde.state.pa.us 2007

M8.A Numbers and Operations M8.A.1 Demonstrate an understanding of numbers, ways of representing numbers, relationships among numbers and number systems. M8.A.1.1 Represent numbers in equivalent forms. Reference: 2.1.8.A, 2.1.8.B M8.A.1.1.1 Represent numbers using scientific notation and/or exponential forms. M8.A.1.1.2 Find the square or cube of a whole number (single digit) and/or the square root of a perfect square (without a calculator) and explain the relationship between the two (i.e. square and square root). The earth is approximately 93,000,000 miles from the sun. What is the distance written in scientific notation? A. 9.3 x 10 6 B. 93 x 10 6 C. 93 x 10 7 D. 9.3 x 10 7 (Maryland Department of Education) 4 Which number represents 4.5 x 10 written in standard notation? A. 0.00045 B. 0.000045 C. 45,000 D. 450,000 8 The number 5.12 x 10 is written in scientific notation. How would it be written in standard form? A. 0.0000000512 B. 51,200,000 C. 512,000,000 D. 51,200,000,000 Liang got an answer of about 3.87 when she entered 15 on her calculator and pressed the ( ) key. As usual, she stopped to think briefly about whether or not her calculator s answer was reasonable. Which of the following statements is the most likely explanation for her to believe that her calculator s answer is or not reasonable? A. It is not reasonable, because the answer should be a whole number. B. It is reasonable because 3 squared is 9 while 4 squared is 16. C. It is not reasonable, because the answer should be only slightly more than 3. D. It is reasonable, because 15 is an odd number. (New Jersey Department of Education) Pennsylvania Department of Education Math Grade 8 - Page 1

M8.A Numbers and Operations M8.A.2 Understand the meanings of operations, use operations and understand how they relate to each other. M8.A.2.1 Complete calculations by applying the order of operations. Reference: 2.2.8.A M8.A.2.1.1 Simplify numeric expressions involving integers, using the order of operations. (May include all types of grouping symbols. No combining negatives with exponents [4-3 ] or compound exponents.) 3 3 + 4(8 5) 6 = A. 6.5 B. 11 C. 27.5 D. 29 E. 34.16 (NAEP) What is the value of the expression: 3 2 + (3 10 3 ) 82 16? A. 7,370 B. 7,372 C. 198,008 D. 245,992 Evaluate 7 + 5[(3 + 2) (2 3 + 1)] 2 A. 22 B. 36 C. 87 D. 97 Pennsylvania Department of Education Math Grade 8 - Page 2

M8.A Numbers and Operations M8.A.2 Understand the meanings of operations, use operations and understand how they relate to each other. M8.A.2.2 Represent or solve problems using rates, ratios, proportions and/or percents. Reference: 2.1.8.D, 2.3.8.B M8.A.2.2.1 Solve problems involving percents (e.g., tax, discounts, etc) Do not include percent increase or decrease. M8.A.2.2.2 Represent or solve rate problems (e.g., unit rates, simple interest, distance, etc.) Students may be asked to solve for any term (formulas provided on the reference sheet for distance and interest). A secretary can type 56 words per minute. How much time will she need to type a 4200-word report? A. 7 hours 30 minutes B. 1 hour 4 minutes C. 1 hour 28 minutes D. 1 hour 15 minutes In 1991, an American, Ann Trason, set a world record by running 100 km in Hours Minutes Seconds 7 50 09 Which is the best estimate of her average speed? A. 12 km per hr B. 14 km per hr C. 16 km per hr D. 18 km per hr (New Jersey Department of Education) A runner ran 3000 m in exactly 8 minutes. What was his average speed in meters per second? A. 3.75 B. 6.25 C. 16.0 D. 37.5 E. 62.5 (TIMSS) Pennsylvania Department of Education Math Grade 8 - Page 3

M8.A Numbers and Operations M8.A.3 Compute accurately and fluently and make reasonable estimates. M8.A.3.1 Determine the appropriateness of overestimating, underestimating or calculating an exact answer in problemsolving situations. Reference: 2.2.8.F M8.A.3.1.1 Identify, use and/or explain when it is appropriate to round up or round down. M8.A.3.1.2 Identify, apply and/or explain when an exact answer is needed or when estimation is appropriate. Pennsylvania Department of Education Math Grade 8 - Page 4

M8.A Numbers and Operations M8.A.3 Compute accurately and fluently and make reasonable estimates. M8.A.3.2 Use estimation strategies in problemsolving situations. Reference: 2.2.8.D M8.A.3.2.1 Estimate answers to problems involving percents (percents will be limited to: 1%, 10%, 15%, 20%, 25%, 50% or 75%). Ken bought a used car for $5,375. He had to pay an additional 15 percent of the purchase price to cover both sales tax and extra fees. Of the following, which is the closest to the total amount Ken paid? A. $806 B. $5,510 C. $5,760 D. $5,940 E. $6,180 (NAEP) A state law requires that students attend school 180 days out of the 365 days in a year. Approximately what percent of a year must students attend school? A. 2% B. 50% C. 75% D. 200% Pennsylvania Department of Education Math Grade 8 - Page 5

M8.A Numbers and Operations M8.A.3 Compute accurately and fluently and make reasonable estimates. M8.A.3.3 Compute and/or explain operations with integers, fractions and/or decimals. Reference: 2.2.8.B M8.A.3.3.1 Add, subtract, multiply and/or divide integers, fractions and/or decimals with and without a calculator (straight computation or word problems). Subtract (-) 14 5-6 8 5 8 A. 8 B. 21 C. 8 D. 7 5 24 11 24 19 24 Divide ( ) 3 0.24 A. 0.08 B. 0.72 C. 12.5 D. 125.00 Subtract (-) 11-1 2 3 A. 9 B. 9 C. 10 D. 10 1 3 2 3 1 3 2 3 Pennsylvania Department of Education Math Grade 8 - Page 6

M8.B Measurement M8.B.1 Demonstrate an understanding of measurable attributes of objects and figures, and the units, systems and processes of measurement. M8.B.1.1 Convert measurements. Reference: 2.3.5.D M8.B.1.1.1 Convert among metric measurements (milli, centi, kilo using meter, liter and gram) (table of equivalency provided on the reference sheet). M8.B.1.1.2 Convert customary measurements up to 2 units above or below the given unit (e.g., inches to yards, pints to gallons) (table of equivalency provided on the reference sheet). M8.B.1.1.3 Convert time up to 2 units above or below given unit (e.g., seconds to hours). M8.B.1.1.4 Convert from Fahrenheit to Celsius or Celsius to Fahrenheit (formulas provided on the reference sheet). 1 mile = 5,280 feet How many feet are in 15 miles? A. 352 B. 35,200 C. 79,200 D. 84,480 E. 89,760 (NAEP) How many yards are equal to 72 inches? A. 2 yards B. 3 yards C. 6 yards D. 36 yards Greg is 150 centimeters tall. How many meters is that? A. 0.500 B. 1.5 C. 15 D. 15,000 (Connecticut State Department of Education) Pennsylvania Department of Education Math Grade 8 - Page 7

M8.B Measurement M8.B.2 Apply appropriate techniques, tools and formulas to determine measurements. M8.B.2.1 Determine the measurement of a missing side(s) or angle(s) in a polygon. Reference: 2.3.8.C, 2.9.8.D M8.B.2.1.1 Determine the total number of degrees in the interior angles of a polygon in 3-8 sided figures (formula provided on the reference sheet). M8.B.2.1.2 Determine the measurement of one interior angle of a regular polygon (3-8 sided polygons, formula provided on the reference sheet). M8.B.2.1.3 Determine the number of sides of a polygon given the total number of degrees in the interior angles (3-8 sided polygons, formula provided on the reference sheet). In a quadrilateral, each of two angles has a measure of 115. If the measure of a third angle is 70, what is the measure of the remaining angle? A. 60 B. 70 C. 130 D. 140 (TIMSS) Find the measure in degrees, of the smallest angle in this triangle? A. 20 B. 40 C. 60 D. 80 Pennsylvania Department of Education Math Grade 8 - Page 8

M8.B Measurement M8.B.2 Apply appropriate techniques, tools and formulas to determine measurements. M8.B.2.2 Use, describe and/or develop procedures to determine measures of perimeter, circumference, area, surface area and/or volume. Reference: 2.3.8.A, 2.3.8.D M8.B.2.2.1 Calculate the surface area of cubes and rectangular prisms (formula provided on the reference sheet). M8.B.2.2.2 Calculate the volume of cubes and rectangular prisms (formulas provided on the reference sheet). M8.B.2.2.3 Determine the appropriate type of measurement (circumference, perimeter, area, surface area, volume) for a given situation (e.g., which measurement is needed to determine the amount of carpeting for a room). The box pictured below is open at the top. Find its outside surface area. 20 cm 10 cm 15 cm A. 45 cm 2 B. 1150 cm 2 C. 1300 cm 2 D. 3750 cm 2 Pennsylvania Department of Education Math Grade 8 - Page 9

M8.C Geometry M8.C.1 Analyze characteristics and properties of two- and three- dimensional geometric shapes and demonstrate understanding of geometric relationships. M8.C.1.1 Identify, use, and/or describe properties of angles, triangles, quadrilaterals, circles, pyramids, cubes, prisms, spheres, cones and/or cylinders. Reference: 2.3.8.C, 2.9.8.B, 2.9.8.E, 2.9.8.D Which angles are complementary? M8.C.1.1.1 Match the three-dimensional figure with its net (cube, cylinder, cone, prism, pyramid). Any measurements used should be consistent in the stem and answer choices. M8.C.1.1.2 Define, identify and/or use properties of angles formed by intersecting lines (complementary, supplementary, adjacent and/or vertical angles). M8.C.1.1.3 Define, identify and/or use properties of angles formed when two parallel lines are cut by a transversal (alternate interior, alternate exterior, vertical corresponding). A. 2 and 3 B. 3 and 4 C. 4 and 5 D. 1 and 2 Pennsylvania Department of Education Math Grade 8 - Page 10

M8.C Geometry M8.C.1 Analyze characteristics and properties of two- and three- dimensional geometric shapes and demonstrate understanding of geometric relationships. M8.C.1.2 Compute measures of sides of right triangles using the Pythagorean Theorem. Reference: 2.10.8.A M8.C.1.2.1 Use the Pythagorean Theorem to find the measure of a missing side of a right triangle (formula provided on the reference sheet whole numbers only). Mr. Kyle drives eight miles south and then six miles east. What was the diagonal distance from his starting point? A. 2 miles B. 10 miles C. 14 miles D. 48 miles What is the length of the missing side in this triangle? A. 14 B. 15 C. 16 D. 18 Pennsylvania Department of Education Math Grade 8 - Page 11

M8.C Geometry M8.C.2 Identify and/or apply concepts of transformations or symmetry. Not assessed at Grade 8. Pennsylvania Department of Education Math Grade 8 - Page 12

M8.C Geometry M8.C.3 Locate points or describe relationships using the coordinate plane. M8.C.3.1 Plot and/or identify ordered pairs on a coordinate plane. Reference: 2.8.5.H M8.C.3.1.1 Plot, locate or identify ordered pairs on a coordinate plane (the point may be a vertex of a polygon). In the figure above, if ABCD is a square, then the coordinates of vertex C are A. (4,5) B. (3,-4) C. (3,-2) D. (5,-4) E. (5,-2) (NAEP) Pennsylvania Department of Education Math Grade 8 - Page 13

M8.D Algebraic Concepts M8.D.1 Demonstrate an understanding of patterns, relations and functions. M8.D.1.1 Analyze, extend or develop descriptions of patterns or functions. Reference: 2.8.8.B, 2.8.8.G, 2.11.8.C The table shows a relation between x and y. x 2 3 4 5 y 7 10 13 16 Which of these equations expresses this relation? A. y = x + 5 B. y = x ± 5 C. y = 3 1 (x ± 1) D. y = 3x + 1 M8.D.1.1.1 Continue a numeric or algebraic pattern (pattern must show 3 repetitions may include up to 2 operations, squares and square roots). M8.D.1.1.2 Find missing elements in numeric or geometric patterns and/or functions (may be given a table or rule pattern must show 3 repetitions). M8.D.1.1.3 Determine the rule of a function (given elements in an input-output table, chart or list limit to linear functions). What is the next number in the pattern? A. 58 B. 64 C. 76 D. 98 4, 13, 28, 49,... List the next two values in this sequence: 4, 10, 22,, Which of the following equations is true for the three pairs of x and y values in the table above? A. 3x + 2 = y B. 3x 2 = y C. 2x + 3 = y D. 2x 3 = y E. x 3 = y (NAEP) A. 34, 46 B. 34, 48 C. 40, 62 D. 46, 94 Pennsylvania Department of Education Math Grade 8 - Page 14

M8.D Algebraic Concepts M8.D.2 Represent and/or analyze mathematical situations using numbers, symbols, words, tables and/or graphs. M8.D.2.1 Select and/or use a strategy to simplify an expression, solve an equation or inequality and/or check the solution for accuracy. Reference: 2.8.8.C, 2.8.8.E If 4x 6 = 14, what is the value of x? M8.D.2.1.1 Solve one- or two-step equations and inequalities (should not include absolute values one variable only). M8.D.2.1.2 Use substitution to check the accuracy of a given value for an equation or inequality (simple inequalities with one variable). M8.D.2.1.3 Determine the value of an algebraic expression by simplifying and/or substituting a number for the variable. A. 1 B. 2 C. 4 D. 5 For which equation is (4, 3) a solution? A. x y = 7 B. x + y = 7 C. x = y = 1 D. x + y = 12 If k can be replaced by any number, how many different values can the expression k + 6 have? A. None B. One C. Six D. Seven E. Infinitely many (NAEP) x 2 < 7 is equivalent to A. x < 2 7 B. x < 5 C. x < 14 D. x > 5 E. x > 14 (TIMSS) Pennsylvania Department of Education Math Grade 8 - Page 15

M8.D Algebraic Concepts M8.D.2 Represent and/or analyze mathematical situations using numbers, symbols, words, tables and/or graphs. M8.D.2.2 Create and/or interpret expressions, equations or inequalities that model problem situations. Reference: 2.8.8.C M8.D.2.2.1 Match a written situation to its numeric and/or algebraic expression, equation or inequality (up to two variables in equations or expressions one variable with inequalities). M8.D.2.2.2 Write and/or solve an equation for a given problem situation (one variable only). Which equation shows that the sum of x and 2 is twice as much as 6? A. x = 2 2 6 B. x + 2 2 = 6 C. 2(x + 2) = 6 D. x + 2 = 2 6 A wooden box with 8 videocassettes inside weighs 4.2 kilograms. The box weighs 0.6 kg when empty. Using w to represent the weight of one videocassette, which of the following describes this situation? A. 8w = 4.2 B. 8w + 0.6 = 4.2 C. 8w 0.6 = 4.2 D. 8(w + 0.6) = 4.2 (New Jersey Department of Education) Which of the following equations gives the rule for finding the numbers in the column on the right? x y 1 7 2 11 3 15 A. y = x + 4 B. y = 2x + 5 C. y = x + 6 D. y = 4x + 3 (New Jersey Department of Education) Pennsylvania Department of Education Math Grade 8 - Page 16

M8.D Algebraic Concepts M8.D.3 Analyze change in various contexts. Not assessed at grade 8. Pennsylvania Department of Education Math Grade 8 - Page 17

M8.D Algebraic Concepts M8.D.4 Describe or use models to represent quantitative relationships. M8.D.4.1 Represent relationships with tables or graphs on the coordinate plane. Reference: 2.8.8.C, 2.8.8.H M8.D.4.1.1 Graph a linear function based on an x/y table (integers only). M8.D.4.1.2 Match the graph of a linear function to its x/y table (integers only). M8.D.4.1.3 Match the linear equation (y = mx + b form) to the x/y table (integers only in the table). Given the function y= 2 1 x 2, which set of numbers completes the table? x -4-2 0 y A. { 4, 3, 2} B. {-4, -3, -2} C. {-4, 3, 2} D. { 4, 3, 2} Pennsylvania Department of Education Math Grade 8 - Page 18

M8.E Data Analysis and Probability M8.E.1 Formulate or answer questions that can be addressed with data and/or organize, display, interpret or analyze data. M8.E.1.1 Choose, display or interpret data (tables, charts, graphs, etc.). Reference: 2.6.5.A, 2.6.8.E, 2.7.8.D, 2.6.3.B According to the graph, what percent of the students chose generic brands? M8.E.1.1.1 Choose and/or explain the correct representation (graph) for a set of data. M8.E.1.1.2 Analyze data and/or answer questions pertaining to data shown in multiple line graphs, circle graphs or histograms. M8.E.1.1.3 Interpret data shown in stem-and-leaf or box-and-whisker plots. The table below shows test scores for a class. How many students scored in the 80 s? A. 15% B. 14% C. 17% D. 16% There are 1,200 students enrolled in Adams Middle School. According to the graph, how many of these students participate in sports? A. 380 B. 456 C. 760 D. 820 E. 1,162 A. 2 students B. 6 students C. 7 students D. 9 students STUDENT PARTICIPATION IN ACTIVITIES AT ADAMS MIDDLE SCHOOL Other 12% Clubs 25% Sports 38% Hobbies 25% (NAEP) Pennsylvania Department of Education Math Grade 8 - Page 19

M8.E Data Analysis and Probability M8.E.2 Select and/or use appropriate statistical methods to analyze data. Not assessed at grade 8. Pennsylvania Department of Education Math Grade 8 - Page 20

M8.E Data Analysis and Probability M8.E.3 Understand and/or apply basic concepts of probability or outcomes. M8.E.3.1 Calculate the probability of an event. Reference: 2.7.8.E M8.E.3.1.1 Find the probability for a mutually exclusive or an independent event (written as a fraction in simplest form). There are 15 girls and 11 boys in a mathematics class. If a student is selected at random to run an errand, what is the probability that a boy will be selected? A. 4 B. 26 11 C. 26 15 26 D. 11 E. 15 15 11 (NAEP) There are 9 packages, 5 red and 4 green. There are calculators inside 4 of the red packages and inside 2 of the green packages. What is the probability of choosing a package containing a calculator from the entire group of packages? A. 4/5 B. 2/3 C. 1/2 D. 4/9 Jan entered a drawing for a dirt bike 5 times. Only 150 entries were received. What is the probability that Jan will win the dirt bike? A. 1/150 B. 1/50 C. 1/30 D. 1/5 Pennsylvania Department of Education Math Grade 8 - Page 21

M8.E Data Analysis and Probability M8.E.3 Understand and/or apply basic concepts of probability or outcomes. M8.E.3.2 Determine the number of combinations and/or permutations for an event. Reference: 2.7.8.A M8.E.3.2.1 Determine/show the number of permutations and/or combinations for an event using up to four choices (e.g., organized list, etc.). Sarah and Tom belong to a soccer league that has 8 teams. Each team will play all of the other teams twice. How many games will be played in all? A. 16 B. 28 C. 56 D. 64 (Colorado Department of Education) Edward conducts a simulation using a coin, a number cube, and a spinner as shown below. What is the number of outcomes for this simulation? A. 3 B. 8 C. 12 D. 48 (Maryland State Department of Education) Pennsylvania Department of Education Math Grade 8 - Page 22

M8.E Data Analysis and Probability M8.E.4 Develop and/or evaluate inferences and predictions or draw conclusions based on data or data displays. M8.E.4.1 Draw conclusions, make inferences and/or evaluate hypotheses based on statistical and data displays. Reference: 2.6.8.C, 2.7.8.E M8.E.4.1.1 Fit a line to a scatter plot and/or describe any correlation between the two variables (positive, negative, strong, weak or none). M8.E.4.1.2 Make predictions based on survey results or graphs (bar, line, circle, scatterplots, etc.). From a batch of 3000 light bulbs, 100 were selected at random and tested. If 5 of the light bulbs in the sample were found to be defective, about how many defective light bulbs would be expected in the entire batch? A. 15 B. 60 C. 150 D. 300 E. 600 The graph shows the time taken for a pendulum to swing backwards and forwards 20 times for different lengths of string. The length of a string is 90 cm. About how long would it take for the pendulum to swing backwards and forwards 20 times? (TIMSS) Fit a line to the scatter plot of two quantities. Describe any correlation of the variables. The data represented in the scatter plot above can be described as having A. Positive correlation B. Negative correlation C. No correlation D. Both positive and negative correlation A. 35 seconds B. 38 seconds C. 42 seconds D. 45 seconds (TIMSS) Pennsylvania Department of Education Math Grade 8 - Page 23